The modified concavity method by Levine is used for proving blow-up of solutions

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

NON-EXISTENCE OF GLOBAL SOLUTIONS TO GENERALIZED DISSIPATIVE KLEIN-GORDON EQUATIONS WITH POSITIVE

ENERGY

MAXIM OLEGOVICH KORPUSOV

Abstract. In this article the initial-boundary-value problem for generalized dissipative high-order equation of Klein-Gordon type is considered. We con- tinue our study of nonlinear hyperbolic equations and systems with arbitrary positive energy. The modified concavity method by Levine is used for proving blow-up of solutions.

1. Introduction We consider the initial-boundary-value problem

u_tt+µu_t+ ∆h₁(x,∆u)−div(h₂(x,|∇u|)∇u) + div(h3(x,|∇u|)∇u) = 0, (1.1) u|∂Ω= ∂u

∂n_x

_∂Ω= 0, u(x,0) =u0(x), u⁰(x,0) =u1(x), µ≥0, (1.2) in a bounded domain Ω⊂R^N with smooth boundary∂Ω∈C^4,δ forδ∈(0,1].

Finite time blow-up of solutions of generalized Klein-Gordon equation have been studied by many authors; see for example [2, 3, 6, 4, 24, 5, 16]. In these ref- erences, the authors considere problems either for negative energy or for weaker conditions than a condition of negative initial energy (see [16, 23]). Other authors have assumed a condition of positive energy under other two conditions on the ini- tial functions. However, the mentioned authors have not studied the compatibility of these conditions, which is come times hard to understand. Finally these condi- tions for any fixed u0 and sufficiently large u1 are not compatible. These authors have used the classic concavity Levine’s method. In this paper we use a modified method, developed in [1], that provide two conditions for which their compatibility is easily checked.

Let us remember that there are five well-known methods for studying a blow- up phenomena. The first method is the concavity method developed by Levine [13, 14, 19, 20, 22, 15]. The second method is the test functions developed by Pokhozhaev, Mitidieri and Zhang [17, 18, 7, 25]. And the third method based on different criterion of comparison and was developed by Samarskii, Galaktionov,

2000Mathematics Subject Classification. 35B44, 35L05, 35L25, 35L67.

Key words and phrases. Blow-up; wave equation; shocks and singularities . c

2012 Texas State University - San Marcos.

Submitted February 14, 2012. Published July 19, 2012.

Supported by grant N 11-01-12018-ofi-m-2011 from the Russian Foundation for basic research.

1

Kurdyumov, Mikhailov [8, 21]. The forth method based of positive averageswas developed by Keller and Glassey [12, 10]. The fifth method, for nonlinear damping, was developed by by Georgiev and Todorova [9].

2. Main differential inequality Consider the important differential inequality

ΦΦ⁰⁰−α(Φ⁰)²+γΦ⁰Φ +βΦ≥0, α >1, β ≥0, γ≥0, (2.1) where Φ(t)∈C⁽²⁾([0, T]), Φ(t)≥0, Φ(0)>0. Dividing both sides (2.1) by Φ^1+α, we obtain

Φ⁰ Φ^α

0

+γΦ⁰

Φ^α +βΦ^−α≥0.

Therefore,

1

1−α(Φ^1−α)⁰⁰+ γ

1−α(Φ^1−α)⁰+βΦ^−α≥0. (2.2) By definition, putZ(t) = Φ^1−α(t). Then, from (2.2) we obtain

Z⁰⁰+γZ⁰−β(α−1)Z^α1 ≤0, α1= α

α−1. (2.3)

Also by definition, putY(t) =e^γtZ(t); hence from (2.3) we obtain Y⁰⁰−γY⁰−β(α−1)e^−δtY^α1 ≤0, δ= γ

α−1. (2.4)

It is easily shown that the following chain of equalities holds:

Y⁰= Φ^1−αe^γt0

= Φ^−α(α−1)e^γt

−Φ⁰(t) + γ α−1Φ(t)

. (2.5)

Take the initial condition

Φ⁰(0)> γ

α−1Φ(0); (2.6)

then there existst₀>0 such that Φ⁰(t)> γ

α−1Φ(t) fort∈[0, t0). (2.7) Combining (2.7) and (2.5), we obtain

Y⁰(t)<0 for t∈[0, t0).

Since−γY⁰(t)≥0, fort∈[0, t0), it follows from (2.4) that Y⁰⁰−β(α−1)e^−δtY^α1 ≤0, δ= γ

α−1 for t∈[0, t0). (2.8) Now multiplying both sides (2.8) byY⁰, we obtain

Y⁰Y⁰⁰−β(α−1)e^−δtY^α1Y⁰ ≥0, δ= γ

α−1 fort∈[0, t0). (2.9) Let us remark that

e^−δtY^α1Y⁰ = d

dt[e^−δtY^1+α1] +δe^−δtY^1+α1−α1e^−δtY^α1Y⁰, Thus we have

e^−δtY^α1Y⁰= 1 1 +α1

d

dt[e^−δtY^1+α1] + 1 1 +α1

δe^−δtY^1+α1. (2.10) Combining (2.10) with (2.9), we obtain

Y⁰Y⁰⁰−β(α−1) 1 +α₁

d

dt[e^−δtY^1+α1]−β(α−1)δ

1 +α₁ e^−δtY^1+α1 ≥0 fort∈[0, t0),

clearly, from this inequality we obtain Y⁰Y⁰⁰−β(α−1)

1 +α1

d

dt[e^−δtY^1+α1]≥0 fort∈[0, t₀). (2.11) Integrating the above expression,

(Y⁰)²≥A²+2β(α−1)²

2α−1 e^−δtY^1+α1 ≥A², (2.12) where

A²≡(Y⁰(0))²−2β(α−1)²

2α−1 Y^1+α1(0). (2.13) We assume the condition

A²>0.

The reader will have no difficulty in showing that this condition is equivalent to the condition

A²= (α−1)²Φ^−2α(0)[ Φ⁰(0)− γ

α−1Φ(0)²

− 2β

2α−1Φ(0)

>0. (2.14) Therefore, the conditionA²>0 and the following condition are equivalent.

Φ⁰(0)− γ

α−1Φ(0)2

> 2β

2α−1Φ(0). (2.15)

Thus, combining (2.12) and (2.14), we obtain Y⁰(t)≤ −A <0⇒Φ⁰(t₀)> γ

α−1Φ(t₀).

But now we have thatY⁰(t0)<0. Therefore, using this algorithm of “continue in time”, we obtain

Y⁰(t)<0 for allt∈[0, T].

This implies that

|Y⁰| ≥A >0⇒Y⁰(t)≤ −A⇒Y(t)≤Y(0)−At⇒

⇒Φ^1−α(t)≤e^−γt[Φ^1−α(0)−At]⇒Φ(t)≥ e^γt/(α−1) [Φ^1−α(0)−At]^1/(α−1). The result is the following theorem.

Theorem 2.1. SupposeΦ(t)∈C⁽²⁾([0, T]), satisfies inequality (2.1)and Φ⁰(0)> γ

α−1Φ(0), (2.16)

Φ⁰(0)− γ

α−1Φ(0)2

> 2β

2α−1Φ(0) (2.17)

whereΦ(t)≥0,Φ(0)>0, then the timeT >0 can not be arbitrarily large, but the following inequality holds

T ≤T_∞≤Φ^1−α(0)A⁻¹, A²≡(α−1)²Φ^−2α(0)

Φ⁰(0)− γ

α−1Φ(0)2

− 2β

2α−1Φ(0) , wherelim sup_t↑TΦ(t) = +∞.

3. Conditions

We begin with conditions on the functionsh1(x, s),h2(x, s), andh3(x, s).

Conditions on h₁(x, s):

(H1.1) h1(x, s) : Ω×R¹→R¹ is a Caratheodory function;

(H1.2) for almost allx∈Ω the functionh1(x, s)∈C⁽¹⁾(R¹) and a “growth condi- tions” take place

|h1(x, s)| ≤c1+c2|s|^p1−1, |h⁰_1s(x, s)| ≤c1+c2|s|^p1−2 forp1≥2; (3.1) (H1.3) for anyv(x)∈W^2,p0 ¹(Ω) there exist the inequalities

0≤ Z

Ω

h1(x,∆v(x))∆v(x)dx≤θ1

Z

Ω

H1(x,∆v(x))dx, (3.2) whereθ1>0 andH1(x, s) =Rs

0 dσ h1(x, σ);

conditions on h2(x, s):

(H2.1) h₂(x, s) : Ω×R¹+→R¹+ is a Caratheodory function;

(H2.2) for almost allx∈Ω the functionh2(x, s)∈C⁽¹⁾(R¹+) and we suppose the following inequalities

0≤h2(x, s)≤c3+c4s^p2−2, |h⁰_2s(x, s)s| ≤c3+c4s^p2−2 forp2≥2; (3.3) (H2.3) for anyv(x)∈W^1,p0 ²(Ω) an inequality holds

0≤ Z

Ω

h2(x,|∇v|)|∇v|²dx≤θ2

Z

Ω

dx H2(x,|∇v|) forθ2>0, (3.4) whereH2(x, s) =Rs

0 dσ h2(x, σ)σ.

Conditions on h3(x, s):

(H3.1) h₃(x, s) : Ω×R¹+→R¹+ is a Caratheodory function;

(H3.2) for almost allx∈Ω the functionh3(x, s)∈C⁽¹⁾(R¹+) and

0≤h3(x, s)≤c5+c6s^p3−2, |h⁰_3s(x, s)s| ≤c5+c6s^p3−2, p3>2; (3.5) (H3.3) for allv(x)∈W^1,p0 ³(Ω) we assume that

Z

Ω

h3(x,|∇v|)|∇v|²dx≥θ3

Z

Ω

dx H3(x,|∇v|) forθ3>2, (3.6) whereH3(x, s) =Rs

0 dσ h3(x, σ)σ.

We define

p^∗=

(N p/(N−p), forN > p;

+∞, forN ≤p.

It can easily be checked that from the conditions on the functionsh1(x, s),h2(x, s), andh3(x, s) we have

∆h₁(x,∆v) :W^2,p0 ¹(Ω)→W^−2,p

0

1(Ω), p⁰₁=p₁/(p₁−1), div(h2(x,|∇v|)∇v) :W^1,p0 ²(Ω)→W^−1,p

0

2(Ω), p2=p2/(p2−1), div(h3(x,|∇v|)∇v) :W^1,p0 ³(Ω)→W^−1,p

0

3(Ω), p3=p3/(p3−1), and this operators are continuous in the corresponding topologies.

Definition 3.1. A strong generalized solution of (1.1), (1.2) is a functionu(x)(t) in the class

u(x)(t)∈C⁽²⁾([0, T];W^2,p0 ¹(Ω)), T >0, for some larges >0, if the following condition hold:

Z

Ω

u⁰⁰(x)(t)w(x)dx+µ Z

Ω

u⁰(x)(t)w(x)dx+ Z

Ω

h1(x,∆u)∆w(x)dx +

Z

Ω

h₂(x,|∇u|)(∇u,∇w)dx− Z

Ω

h₃(x,|∇u|)(∇u,∇w)dx= 0, t∈[0, T] (3.7)

for allw(x)∈W^2,p0 ¹(Ω); and

u(x)(0) =u0(x)∈W^2,p0 ¹(Ω), u⁰(x)(0) =u1(x)∈L²(Ω). (3.8) 4. Blow-up of solutions

Assume that there is a weak solutionu(x)(t) in the classC⁽²⁾([0, T];W^2,p0 ¹(Ω)) for someT >0. Let us putw=u(x)(t) in equation (3.7), then we obtain the first energy equality

1 2

d²Φ dt² +µ

2 dΦ

dt −J+ Z

Ω

h1(x,∆u)∆u dx+ Z

Ω

h2(x,|∇u|)|∇u|²dx

= Z

Ω

h₃(x,|∇u|)|∇u|²dx,

(4.1)

where we denote

Φ(t)≡ Z

Ω

|u|²dx, J(t)≡ Z

Ω

|u⁰|²dx.

Let us putw=u⁰(x)(t) in (3.7), we obtain the second energy equality d

dt 1

2J+ Z

Ω

H1(x,∆u)dx+ Z

Ω

H2(x,|∇u|)dx +µJ

= d dt

Z

Ω

H3(x,|∇u|)dx.

(4.2)

Furthermore, integrating (4.2) over time, we obtain the inequality 1

2J+ Z

Ω

H1(x,∆u)dx+ Z

Ω

H2(x,|∇u|)dx−E(0)≤ Z

Ω

H3(x,|∇u|)dx, (4.3) where

E(0)≡ 1 2 Z

Ω

|u1|²dx+ Z

Ω

H₁(x,∆u₀)dx +

Z

Ω

H₂(x,|∇u0|)dx− Z

Ω

H₃(x,|∇u0|)dx.

(4.4)

By (4.3) we obtain the inequality θ3

2 J+θ3

Z

Ω

H1(x,∆u)dx+θ3

Z

Ω

H2(x,|∇u|)dx−θ3E(0)

≤θ3

Z

Ω

H3(x,|∇u|)dx,

combining this with the condition (H3.3), we obtain θ3

2 J+θ3

Z

Ω

H1(x,∆u)dx+θ3

Z

Ω

H2(x,|∇u|)dx−θ3E(0)

≤ Z

Ω

h3(x,|∇u|)|∇u|²dx.

Using the above inequality and (4.1), we obtain 1

2 d²Φ

dt² +µ 2

dΦ dt −J+

Z

Ω

h1(x,∆u)∆u dx+ Z

Ω

h2(x,|∇u|)|∇u|²dx

≥ θ3

2J +θ3

Z

Ω

H1(x,∆u)dx+θ3

Z

Ω

H2(x,|∇u|)dx−θ3E(0).

(4.5)

Now taking into account the conditions (H3.1) and (H3.2), from (4.5) we obtain 1

2 d²Φ

dt² +µ 2

dΦ

dt −J+θ₁ Z

Ω

H₁(x,∆u)dx+θ₂ Z

Ω

H₂(x,|∇u|)dx

≥ θ₃ 2J+θ3

Z

Ω

H1(x,∆u)dx+θ3

Z

Ω

H2(x,|∇u|)dx−θ3E(0).

(4.6)

Under the conditionsθ₃≥θ₁,θ₃≥θ₂using the inequalities Z

Ω

H1(x,∆u)dx≥0, Z

Ω

H2(x,|∇u|)dx≥0, from (4.6), we obtain

1 2

d²Φ dt² +µ

2 dΦ

dt +θ₃E(0)≥ 1 +θ₃ 2

J(t). (4.7)

Using the Cauchy-Bunyakovsky-Schwarz inequality, it is easily shown the differen- tial inequality

(Φ⁰)²≤4JΦ. (4.8)

Combining (4.7) and (4.8), we obtain the important differential inequality ΦΦ⁰⁰−1

2 1 + θ3

2

(Φ⁰)²+µΦΦ⁰+ 2θ₃E(0)Φ≥0. (4.9) Comparing this differential inequality with (2.1), we obtain that

α=1 2 1 +θ3

2

>1 forθ3>2, β= 2θ₃E(0), γ=µ, 2β

2α−1 = 8E(0), γ

α−1 = 4µ θ3−2. We assume the following conditions

Φ⁰(0)> 4µ

θ3−2Φ(0)>0, (4.10)

Φ⁰(0)− 4µ

θ3−2Φ(0)²

>8E(0)Φ(0), (4.11)

E(0)≡1 2

Z

Ω

|u1|²dx+ Z

Ω

H₁(x,∆u₀)dx +

Z

Ω

H₂(x,|∇u₀|)dx− Z

Ω

H₃(x,|∇u₀|)dx >0,

(4.12)

Under conditions (4.10)–(4.12), the time T >0 of existence ofu(x)(t) is bounded from above

T ≤Φ^(2−θ3)/4(0)A⁻¹, A²≡ θ3−2

4 ²

Φ^−1−θ3/2(0)h

Φ⁰(0)− 4µ

θ₃−2Φ(0)2

−8E(0)Φ(0)i ,

at the same time

Φ(t)≥ e^{4µt/(θ3−2)}

[Φ^(2−θ3)/4(0)−At]^4/(θ3−2), (4.13) where

Φ⁰(0) = 2 Z

Ω

u1(x)u0(x)dx, Φ(0) = Z

Ω

|u0|²dx.

Therefore, our main result of is the following theorem.

Theorem 4.1. Assume all conditions onh₁,h₂ andh₃ hold. Under the following conditions

Φ⁰(0)> 4µ

θ3−2Φ(0) + (8E(0)Φ(0))^1/2>0, E(0)>0, (4.14)

θ3≥θ1, θ3≥θ2, (4.15)

there exists the estimation from above for the time of solution existenceT, T1≤T_∞= Φ^(2−θ3)/4(0)A⁻¹;

i. e., we have that lim sup_t↑T1Φ(t) = +∞, where Φ(0) =

Z

Ω

|u0|²dx, Φ⁰(0) = 2 Z

Ω

u1u0dx, E(0)≡ 1

2 Z

Ω

|u1|²dx+ Z

Ω

H1(x,∆u0)dx +

Z

Ω

H2(x,|∇u0|)dx− Z

Ω

H3(x,|∇u0|)dx, A²≡ θ3−2

4 2

Φ^−1−θ3/2(0)

Φ⁰(0)− 4µ

θ3−2Φ(0)2

−8E(0)Φ(0) .

Remark 4.2. Now we shall see that all conditions (4.14) are compatible for enough small µ ≥0. Indeed, first we shall chooseu₀ ∈ W^2,p0 ¹(Ω) enough large to satisfy the inequality

Z

Ω

H3(x,|∇u0|)dx

>

Z

Ω

H1(x,∆u0)dx+ Z

Ω

H2(x,|∇u0|)dx+ 2 θ3−2

Z

Ω

|u0|²dx.

(4.16)

Secondly we fixu0 and choose u1=λu0 forλ >2µ/(θ3−2). In this case we have Φ⁰(0)− 4µ

θ3−2Φ(0) = 2

λ− 2µ θ3−2

Φ(0)>0. (4.17) Finally we chooseλ >2µ/(θ₃−2) enough large to satisfy the inequality

E(0) = λ² 2

Z

Ω

|u0|²dx+ Z

Ω

H1(x,∆u0)dx +

Z

Ω

H2(x,|∇u0|)dx− Z

Ω

H3(x,|∇u0|)dx >0.

Under condition (4.17), inequality (4.14) is equivalent to

Φ⁰(0)− 4µ

θ₃−2Φ(0)2

>8E(0)Φ(0), (4.18)

by our substitution we obtain from the left and right side of this inequality

Φ⁰(0)− 4µ

θ₃−2Φ(0)2

= 4

λ− 2µ θ₃−2

2

Φ²(0)

=

4λ²− 16µλ

θ₃−2 + 16µ² (θ₃−2)²

Φ²(0),

(4.19)

and

8E(0)Φ(0) = 4λ²Φ²(0) + 8Φ(0)hZ

Ω

H1(x,∆u0)dx +

Z

Ω

H₂(x,|∇u0|)dx− Z

Ω

H₃(x,|∇u0|)dxi (4.20) Combining (4.18) with (4.19) and (4.20), we obtain that

Φ⁰(0)− 4µ

θ₃−2Φ(0)2

=

4λ²− 16µλ

θ₃−2+ 16µ² (θ₃−2)²

Φ²(0)>8E(0)Φ(0)

= 4λ²Φ²(0) + 8Φ(0)hZ

Ω

H₁(x,∆u₀)dx +

Z

Ω

H₂(x,|∇u0|)dx− Z

Ω

H₃(x,|∇u0|)dxi ,

(4.21)

Now it is not hard to prove that Z

Ω

H₃(x,|∇u0|)dx+ 2µ² (θ3−2)²

Z

Ω

|u0|²dx

> 2µλ θ3−2

Z

Ω

|u0|²dx+ Z

Ω

H1(x,∆u0)dx+ Z

Ω

H2(x,|∇u0|)dx.

(4.22)

Moreover, we choose

λ= 1

µ forµ∈ 0,(θ3−2 2 )^1/2

,

and ifµ= 0, thenλ >0 and large enough. We see that all foregoing conditions are satisfied for small enoughµ ≥0. Now combining this large enoughλ and (4.22), we obtain the inequality

Z

Ω

H₃(x,|∇u0|)dx+ 2µ² (θ3−2)²

Z

Ω

|u0|²dx

> 2 θ3−2

Z

Ω

|u0|²dx+ Z

Ω

H1(x,∆u0)dx+ Z

Ω

H2(x,|∇u0|)dx.

Obviously, this inequality holds by (4.16). Therefore, we have to prove (4.16) for some functions onh1(x, s),h2(x, s), and h3(x, s). At the same time we check the condition (4.15). Suppose

h1(x, s) =|s|^p1−2s, h2(x, s) =s^p2−2, h3(x, s) =s^p3−2, wherep3> p1>2,p3> p2≥2. Then

H₁(x, s) =|s|^p1 p1

, H₂(x, s) =|s|^p2 p2

, H₃(x, s) =|s|^p3 p3

,

and θ₃ = p₃ > θ₁ = p₁ > 2, θ₃ = p₃ > θ₂ = p₂. Therefore, first note that the condition (4.15) holds, and further note that for large enoughu0(x)∈W^2,p0 ¹(Ω) the condition (4.16) also holds.

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Maxim Olegovich Korpusov

Chair of Mathematics, Faculty of Physics, M. V. Lomonosov Moscow State University, Moscow, Russia

E-mail address:[email protected]